The adjoint T*superscriptTT^{*} of a linear transformationTTT is linear transformation such that ⟨T⁢x,y⟩=⟨x,T*⁢y⟩TxyxsuperscriptTy\langle Tx,y\rangle=\langle x,T^{*}y\rangle, for any pair of vectors x,y∈VxyVx,y\in V.

If VVV is non-singular with respect to the inner product ⟨⋅,⋅⟩fragmentsnormal-⟨normal-⋅normal-,normal-⋅normal-⟩\langle\cdot,\cdot\rangle and that the adjoint T*superscriptTT^{*} of a linear transformation TTT exists, it is not hard to show that

TTT is an isometry if and only if T⁢T*=I=T*⁢TTsuperscriptTIsuperscriptTTTT^{*}=I=T^{*}T.

More generally, in a ring with involution**, an isometry (or an unitary element) is a unit (both a left unit and a right unit) aaa whose product with its adjoint a*superscriptaa^{*} is 1 (i.e. its inverse is its adjoint). Now, if aaa is not a unit, this product a⁢a*asuperscriptaaa^{*}
will not be 1. The next best thing to hope for is that the product will be an idempotent. But because a⁢a*asuperscriptaaa^{*} is self-adjoint, this idempotent is in fact a projection. This is how a partial isometry is defined. Formally,

let RRR be a ring with involution **, an element a∈RaRa\in R is a partial isometry if a⁢a*asuperscriptaaa^{*} and a*⁢asuperscriptaaa^{*}a are both projections.

Given a partial isometry aaa, the projections a*⁢asuperscriptaaa^{*}a and a⁢a*asuperscriptaaa^{*} are respectively called the initial projection and final projection of aaa.

Examples. Under this definition, 000 is a partial isometry, and so is any isometry.

Remark. If the ring RRR is a Baer *-ring, an element aaa is a partial isometry iffa⁢a*⁢a=aasuperscriptaaaaa^{*}a=a (so a*⁢a⁢a*=a*superscriptaasuperscriptasuperscriptaa^{*}aa^{*}=a^{*}; aaa and a*superscriptaa^{*} are generalized inverses of one another).